A367937 Expansion of e.g.f. exp(4*(exp(x) - 1) + 3*x).

1, 7, 53, 431, 3741, 34471, 335621, 3438943, 36954285, 415187415, 4864054165, 59278367247, 749926582717, 9829744447495, 133267495918885, 1865916660838847, 26942271261464525, 400673643394972983, 6129834703935247285, 96368617886967750767, 1555302323744129219293

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f. A(x) satisfies: A(x) = 1 + x * ( 3 * A(x) + 4 * A(x/(1 - x)) / (1 - x) ).

a(n) = exp(-4) * Sum_{k>=0} 4^k * (k+3)^n / k!.

a(0) = 1; a(n) = 3 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

Mathematica

nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 07 2023

Crossrefs

Cf. A005494, A035009, A078945, A355252, A366199, A367889.

Sequence in context: A133588 A163306 A081338 * A142981 A350252 A120272

Adjacent sequences: A367934 A367935 A367936 * A367938 A367939 A367940

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 05 2023

Status

approved